Q:

Thetime(inhours)requiredtorepairamachineis an exponentially distributed random variable with parameter λ = 1 . What is 2 (a) the probability that a repair time exceeds 2 hours? (b) the conditional probability that a repair takes at least 10 hours, given that its duration exceeds 9 hours?

Accepted Solution

A:
Answer:a) 0.1353b) 0.3679Step-by-step explanation:Let's start by defining the random variable T.T : ''The time (in hours) required to repair a machine''T ~ exp (λ)T ~ exp (1)The probability density function for the exponential distribution is (In the equation I replaced λ = L)[tex]f(x)=Le^{-Lx}[/tex]With L > 0 and x ≥ 0In this exercise λ = 1 ⇒[tex]f(x)=e^{-x}[/tex]For a)[tex]P(T>2)[/tex][tex]P(T>2)=1-P(T\leq 2)[/tex][tex]P(T>2)=1-\int\limits^2_0 {e^{-x} } \, dx[/tex][tex]P(T>2)=1-(-e^{-2}+1)[/tex][tex]P(T>2)=e^{-2}=0.1353[/tex]For b)[tex]P(T\geq 10/T>9)[/tex]The event (T ≥ 10 / T > 9) is equivalent to the event T ≥ 1 so they have the same probability of occur[tex]P(T\geq 10/T>9)=P(T\geq 1)[/tex][tex]P(T\geq 1)=1-P(T<1)=1-\int\limits^1_0 {e^{-x} } \, dx[/tex][tex]P(T\geq 1)=1-(-e^{-1}+1)=e^{-1}=0.3679[/tex]